Numerical Solution of Time and Space Fractional Burger's Equation with Finite Difference Method

Abstract

In this study, fractional Burger’s Equation, which has Dirichlet Boundary Conditions, is solved with the Finite Difference Method. Fractional Burger Equation is found by S. Momani, which is made with changing time and space terms with fractional terms. This equation is solved with the finite difference method and analysis of this scheme is discussed with examples. Stability and Uniqueness are discussed with using matrix method. We compare analytical and numerical solutions with error analysis of them.

Keywords

• Finite difference method,Burger’s equation,Fractional derivative

References

1. [1] Zhang, Y., “A finite difference method for fractional partial differential equation”, Applied Mathematics and Computation 215 (2009) : 524-529.
2. [2] Podlubny, I., “Fractional differential equation”, V198 of Mathematics in Science and Engineering, Academic Press, San Diego, (1999).
3. [3] Zhang, P.G., Wang, J.P., “A predictor–corrector compact finite difference scheme for Burgers’ equation”, Applied Mathematics and Computation 219 (2012) : 892-898.
4. [4] Varöglu, E., Liam Finn, W.D., “Space-time finite elements incorporating characteristics for the burgers’ equation”, Int. J. Numer. Methods Eng. 16 (1980) : 171–184.
5. [5] Kutluay, S., Bahadir, A.R., Özdes, A., “Numerical solution of one-dimensional burgers equation: explicit and exact-explicit finite difference methods”, J. Comput. Appl. Math 103 (1999) : 251–261.
6. [6] Pandey, K., Verma, L.,. Verma, A.K, “On a finite difference scheme for burgers’ equation”, Appl. Math. Comput 215 (2009) : 2206-2214.
7. [7] Liao, W., “An implicit fourth-order compact finite difference scheme for one-dimensional burgers’ equation”, Appl. Math. Comput 206 (2008) : 755-764.
8. [8] Saria, M., Gürarslanb, G., “A sixth-order compact finite difference scheme to the numerical solutions of burgers’ equation”, Appl. Math. Comput 208 (2009) : 475-483.

9. [9] Hon, Y., Mao, X., “An efficient numerical scheme for burgers’ equation”, Appl. Math. Comput 95 (1998) : 37-50.
10. [10] Asaithambi, A., “Numerical solution of the burgers’ equation by automatic differentiation”, Appl. Math. Comput 216 (2010) : 2700-2708.
11. [11] Mena, S.A., Salim, B.C., “Ettahlil-lül adedil muadeleti burger biistihdamil frukatil müntahiyeti”, Mecelle-tül rafdin liulumil muhasebeti verriyaziyatil mücellidi, (2007) : 1-4.
12. [12] Burgers, J.M., “A mathematical model illustrating the theory of turbulence”, Adv. Appl. Mech 1 (1948) : 171-199.
13. [13] Kurulay, M., Bayram, M., “Some properties of the Mittag-Leffler functions and their relation with the Wright functions”, Advances in Difference Equations (2012) : 181.
14. [14] Kurulay, M., “The approximate and exact solutions of the space- and time-fractional Burgers equations”, International Journal of Research and Reviews in Applied Sciences, 3 (2010) : 257-263.
15. [15] Momani, S., “Non-perturbative analytical solutions of the space and time fractional Burgers equations”, Chaos, Solutions and Fractals, 28 (4) (2006) : 930-937.
16. [16] Bashan, A., Karakoc, S.B.G., Geyikli, T., “Approximation of the KdVB equation by the quintic B-spline differential quadrature method”, Kuwait journal of science, 42 (2) (2015) : 67-92.
17. [17] Yılmaz, S., Ünlütürk Y., Mağden, A., “A study on the characterizations of non-null curves according to the Bishop frame of type-2 in Minkowski 3-space”, SAÜ Fen Bil Der 20 (2) (2016) : 325-335.
18. [18] Meerschaert, M.M., Tadjeran, C., “Finite difference approximations for fractional advection–dispersion flow equations”, J. Comput. Appl. Math. 172 (2003) : 65-77.
19. [19] Meerschaert, M.M., Scheffler, H.P., Tadjeran, C., “Finite difference method for two dimensional fractional dispersion equation”, J. Comput. Phys. 211 (2006) : 249–261.